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Defects and degeneracies in supersymmetry protected phases
Fokkema, T.; Schoutens, K.
DOI
10.1209/0295-5075/111/30007
Publication date
2015
Document Version
Accepted author manuscript
Published in
Europhysics Letters
Link to publication
Citation for published version (APA):
Fokkema, T., & Schoutens, K. (2015). Defects and degeneracies in supersymmetry protected
phases. Europhysics Letters, 111(3), [30007]. https://doi.org/10.1209/0295-5075/111/30007
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Defects and degeneracies in supersymmetry protected phases
Thessa Fokkema1∗ and Kareljan Schoutens1,2†
1 Institute for Theoretical Physics, University of Amsterdam Science Park 904, 1098 XH Amsterdam 2 Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP
(Dated: v3, October 8, 2015)
We analyse a class of 1D lattice models, known as Mkmodels, which are characterised by an
order-k clustering of spin-less fermions and by N = 2 lattice supersymmetry. Our main result is the identification of a class of (bulk or edge) defects, that are in one-to-one correspondence with so-called spin fields in a corresponding Zk parafermion CFT. In the gapped regime, injecting such
defects leads to ground state degeneracies that are protected by the supersymmetry. The defects, which are closely analogous to quasi-holes over the fermionic Read-Rezayi quantum Hall states, display characteristic fusion rules, which are of Ising type for k = 2 and of Fibonacci type for k = 3.
PACS numbers: 05.30.-d, 71.10.Fd
Introduction. - In the field of topological quantum
computation (TQC) [1], a number of important lessons have been learned. The first is that non-Abelian statistics tend to be associated to a form of pairing or clustering in a quantum condensate. In 2D, p-wave pairing of spin-less fermions in a p + ip superconductor or Moore-Read (MR) quantum Hall state gives rise to non-Abelian statis-tics of Ising type, through the mechanism of Majorana bound states at the cores of half flux quantum vortices or quasi-holes [2, 3]. In the quantum Hall context, going beyond Ising anyons requires going beyond pairing, as in
the Read-Rezayi (RRk) [4] and NASSk states [5]. The
simplest examples beyond the MR state, the RR3 and
NASS2 states, both give rise to Fibonacci anyons, which
are universal for TQC (see, for example, [6]). More
gen-erally, the anyons carried by the RRk states are universal
for k = 3 and k ≥ 5 [7, 8].
A second lesson learned is that a TQC-through-braiding protocol can be defined not just in 2D but also in a 1D setting [9]. One starts from a T-shaped wire junction with non-Abelian defects at the wire ends and then runs a protocol of braiding, either in position space (by moving defects along the wires) or in parame-ter space. Again the prototypical example are Majorana bound states at the defect points. The underlying pair-ing is typically assumed to be extrinsic, meanpair-ing that it is induced (as in the Kitaev chain) through the proximity of a nearby superconductor.
We here consider the question if one can construct 1D lattice models with built-in, intrinsic, pairing or clus-tering properties and with defects binding Majorana or parafermion zero modes. We find that this goal is
achieved by the lattice models Mk introduced in [10] and
further analysed here. The definition of the Mk models
involves a hard-wired k-clustering constraint as well as N = 2 supersymmetry. The order-k clustering leads to
Zk parafermion degrees of freedom in the CFT
describ-ing the Mk models at criticality. In fact, the supercharge
operator, which injects an extra particle into the
sys-tem, contains the parafermion field ψ1. A similar
struc-ture is maintained if we move into a gapped phase. Our main result is the identification of a class of (bulk and
edge) Mk model defects that precisely correspond to the
so-called spin fields σi in the parafermion CFT. These
defects are in many ways analogous to the quasi-holes
over the RRk quantum Hall states: they have fractional
particle number and display characteristic non-Abelian fusion rules. The underlying mechanism is that of su-persymmetry protected order that is in essence of charge density wave (CDW) type. To turn this into supersym-metry protected topological order in the 1D sense will require a non-local reformulation via a Jordan-Wigner type transformation.
Mk models. - The Mk lattice models [10] describe
spin-less fermions on a 1D lattice, subject to the ‘order-k clustering’ constraint that at the most ‘order-k particles can
occupy consecutive sites. A supercharge Q+is defined as
Q+= L X j=1 X a,b λ[a,b],jd†[a,b],j, (1)
where d†[a,b],j is a fermionic creation operator which
cre-ates a particle at lattice site j in such a way that a string of a particles is formed, with the newly created particle
at position b. Choosing the λ[a,b],j such that (Q+)2= 0,
we define a N = 2 supersymmetric hamiltonian through
H = {Q+, Q−}, (2)
with Q− = (Q+)†. This hamiltonian combines hopping
terms with local potential and interaction terms. By
con-struction, [H, Q+] = [H, Q−] = 0. All states in the
spec-trum are doublets with [f, f + 1] particles, with the ex-ception of the supersymmetric groundstates at E = 0,
which are annihilated by both Q+ and Q−.
Possible choices for the λ[a,b],j have been studied in
[10, 11]. Here we choose for the M2model
λ[1,1],j=
√
2 λj, λ[2,1],j= λ[2,2],j = λj, (3)
2
with the λj staggered as . . . 1λ1λ1 . . .. The factor
√ 2 guarantees that the model is integrable [11] and, if λ = 1, critical [10].
For the general Mk model we choose parameters
de-scribing a critical point perturbed by a specific, inte-grable, staggering [12]. The staggering, with lattice pe-riodicity k + 2, connects the critical regime with one of
‘extreme staggering’ λ 1, where the λ[a,b],j follow a
simple pattern. For k = 3, to lowest order in λ,
λ[1,1],j: . . . 1 √2 √2λ √2 1 . . . λ[2,1],j: . . . 1 1 λ √ 2 λ . . . λ[2,2],j: . . . λ √ 2 λ 1 1 . . . λ[3,1],j: . . . 1 λ λ 1 λ/ √ 2 . . . λ[3,2],j: . . . λ/ √ 2 1 λ2/√2 1 λ/√2 . . . λ[3,3],j: . . . λ/ √ 2 1 λ λ 1 . . . (4)
with the dots indicating repetition modulo 5. We denote this as . . . ? ?λ ? ? . . ., with the ‘λ’ indicating the central position in the staggering pattern.
The Witten index for the Mk model with periodic
boundary conditions (PBC) and with L = l(k + 2) sites
is Wk = k + 1; indeed, for λ > 0 the models have
pre-cisely this number of supersymmetric groundstates, all at E = 0 and filling ν = k/(k + 2) [10, 13]. They are pro-tected against perturbations that commute with super-symmetry and do not affect the k-clustering constraints. For open BC there are either zero or a single supersym-metric groundstate with E = 0, the latter for L ≡ 0, −1 mod (k + 2). We find, however, that in the presence of suitable boundary or bulk defects, the open systems have states with energies that are exponentially suppressed,
E ∝ e−αLi, with L
i characteristic distances among
de-fects and boundaries. The exponential degeneracies are protected by supersymmetry.
M2 model. - We now zoom in on the M2 model on
an open chain. At criticality (λ = 1), the finite size spectra can be matched with those of the 2nd minimal model of N = 2 superconformal field theory (CFT), of
central charge c = 3
2. The match can be made with
the help of numerical spectra (we analysed open chain spectra up to length L = 25, fig. 1) and are similar to
the results of [14] for the M1 model. The relevant CFT
modules are Vm, ψVm with m ∈ Z + 12 and σVm with
m ∈ Z. Here the Vm are charge m vertex operators for
a c = 1 scalar field and the ψ, σ arise from the c = 1
2
Ising CFT factor. States in Vm have an even number
of ψ-modes while those in ψVmcontain an odd number.
The supercharges are the (Ramond sector) zero modes of
the supercurrents ψV±2(z).
Up to an overall 1/L scaling, the lattice model energies
correspond to ECFT= L0−161. The lowest energies are
(4m2− 1)/16 for V
m, (4m2+ 7)/16 for ψVm and m2/4
for σVm. The lowest-energy states in V±1
2 and σV0 are
supersymmetry singlets with ECFT = 0, all other states
have ECFT > 0 and pair up into doublets. The critical
M2 spectrum with open BC is easily described: with
m = 2f − L − 1
2, one finds the CFT modules Vm for
f even and ψVmfor f odd (fig. 1).
Reducing λ below 1 sends the theory off criticality, with RG flow leading to the supersymmetric sine-Gordon
(ssG) theory at coupling β2= 8π [11]. The M
2model
off-critical finite size spectra can be analysed in terms of ssG bulk S-matrices and boundary reflection matrices [15]. The ssG theory holds important clues for the topological
aspects of the M2 model degeneracies [15–17].
Rather than following the RG flow, we will here con-sider the limit λ 1 (‘extreme staggering’), where the
M2eigenstates approach a simple factorized form. This is
analogous to a special tuning in the Kitaev chain, which leads to perfectly decoupled Majorana edge states [9]. This simple setting enables us to demonstrate how dif-ferent BC result in exponential ground state degeneracy beyond this idealised limit. The λ 1 limit is also
sim-ilar to the thin-torus limit of the MR and RRk quantum
Hall states [18–20]. Indeed, the systematics of the fusion channel degeneracies is highly analogous between the two settings.
For λ = 0 and PBC, the M2groundstates are
|−i = . . . 0(·1·)0(·1 . . . , |+i = . . . 1·)0(·1·)0 . . . ,
|0i = . . . 1010101 . . . , (5)
where (·1·) = 110 + 011 and |−i and |+i are related by a shift over two lattice sites. For open BC, there is at the most a single E = 0 groundstate for given particle number f . For L = 4l − 1, staggering 1λ . . . λ1, f = 2l,
|+io,o = [(·1·)0(·1·) . . . (·1·)] , (6)
where ‘o,o’ refers to open/open BC. For λ > 0 this state remains at E = 0, where it is protected by the Witten index, W = 1, and it is separated from all other states by a gap that remains finite as long as λ < 1.
Boundary defects. -To steer into a case with
expo-nentially degenerate groundstates at given particle num-ber f , we need to enforce a defect at both boundaries that allows all three λ = 0 PBC groundstates to connect to the edge at zero energy cost. For this we impose the constraint that the two sites adjacent to a boundary can-not both be occupied by a particle [21]. With this BC
(which we call of ‘σ-type’ and denote by a bracket . . .]σ),
all three λ = 0 PBC vacua can connect to the boundary at zero energy cost. This gives, for L = 4l + 1, λ1 . . . 1λ,
|−iσ,σ=σ[0(·1·) . . . 0]σ, |+iσ,σ =σ[100(·1·) . . . 001]σ,
|0iσ,σ=σ[1010 . . . 0101]σ. (7)
For λ > 0 the state |−iσ,σ at f = 2l remains at E = 0
while |+iσ,σ at f = 2l and |0iσ,σ at f = 2l + 1 pair into
−7/2 −3/2 1/2 5/2 0 1 2 3 4 ψV1 2 V−3 2 V5 2 ψV−7 2 4 } 3 5 m E (a) BC open/open −3 −1 1 3 0 1 2 3 4 σV−1 } 2 } 4 8 σV1 } 2 4 σV3 σV−3 m E (b) BC σ/open −5/2 −1/2 3/2 7/2 0 1 2 3 4 V−1 2 ψV−1 2 V32 ψV3 2 V−5 2 ψV−5 2 V7 2 ψV7 2 m E (c) BC σ/σ
FIG. 1. Numerical M2spectra with L = 25, f = 11, 12, 13, 14 up to E = 4. The labels specify the corresponding CFT modules.
between |±iσ,σ originates from the boundary, which
im-plies that it involves a power of λ that scales with the length of the system. We checked numerically that
δE(λ) ∝ λ(L−1)/2, (8)
which gives exponential splitting δE ∝ e−αL as long as
λ < 1. We checked that this behaviour is robust against perturbations obtained by deforming some of the
param-eters λ[a,b],j in eq. (1), provided we do not break the
N = 2 supersymmetry. We thus identify supersymme-try as the agent protecting the exponential degeneracy
of our ‘qubit’ |±iσ,σ. We remark that many ‘natural’
perturbations of the M2 model do break supersymmetry.
An example is the local fermion density operator ρi at
site i = 4p + 1. At λ = 0 the expected value is hρii = 0
in the state |−iσ,σ while hρii = 1 in the state |+iσ,σ.
Following the ‘qubit’ states all the way to the CFT point, λ = 1, we find (fig. 2)
|−iσ,σ↔ |V−1
2i, |+iσ,σ↔ |ψV−12i, (9)
where |·i denotes the lowest weight state of the corre-sponding CFT module. At the CFT point the boundary Majorana modes have delocalised and the energy split-ting is of order 1/L.
The degeneracy of the ‘qubit’ |±iσ,σ can be traced to
the fusion channel degeneracy σσ = 1 + ψ of the Ising spin-field σ in the underlying CFT. Through the qH-CFT connection [3], this choice of fusion channel carries over to the fusion product of two quasi-holes over the MR state. We consider the MR state in spherical geometry, which we view as an open ‘tube’ capped by specific boundary conditions at the two poles. For N = 2l particles, the MR groundstate has the following thin-torus form
MR, N = 2l : [11001100 . . . 110011] . (10)
The analogous groundstate of the M2 model is precisely
the state |+io,o in eq. (6). The simplest case with
two-fold fusion channel degeneracy is that of the MR states
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 1.5 2 2.5 3 λ E
FIG. 2. Flow between ‘extreme staggering’ (left, λ = 0) and critical (right, λ = 1) limits of the M2 model with L = 17
sites, f = 8 particles, staggering λ1 . . . 1λ and σ/σ BC. The lower two states constitute the ‘qubit’ eq. (9).
with n = 4 quasi-holes. The general counting formula for n quasi-holes reads [22] X F ≡N mod 2 N −F 2 + n n n/2 F . (11)
Here the first binomial counts orbital degeneracies of the n quasi-holes, while the second, together with the sum over F , pertains to the fusion channel degeneracy. We fix the orbital degeneracy by selecting the states with two quasi-holes at both the north and the south poles,
F = 0 : [01100 . . . 110], F = 2 : [10011 . . . 001]. (12)
These ‘MR qubit’ states correspond to the ‘M2 qubit’
states |±iσ,σ of eq. (7). To understand this we have to
compare the open M2 chains with the MR states not on
the sphere but on the cylinder. On a cylinder with vacua ‘1100’ at far left and right, we can extend the F = 0 state
4 as
. . . 1100σσ[01100 . . . 110]σσ0011 . . . , (13)
where σσ denote the two quasi-holes at the boundaries. We can move one of the quasi-holes out from each of the boundaries to get
. . . 1100σ1010 . . . 10σ[01100 . . . 110]σ0101 . . . 01σ0011 . . . .
(14) This corresponds to the situation that we have in the
M2 model, where σ-type BC arise from the presence of
a single σ quantum at a boundary. This interpretation is confirmed by the CFT content of the open chain finite size spectra at λ = 1 (fig. 1). For open/σ BC, we find
all sectors σVm with m shifted to m = 2f − L. For σ/σ
BC, with non-Abelian defects at both ends, we find, at
m = 2f − L + 1
2, both the Vm and ψVm modules, in
accordance with the fusion rule σσ = 1 + ψ. The states in eq. (9) are a particular example.
Bulk defects and quantum register. -At extreme
staggering, kinks connecting any two of the λ = 0 vacua come at finite energy cost. For example, kinks/anti-kinks connecting |0i and |+i, written as
. . . 10101
K
0 0(·1·) . . . , . . . 10101
K
1 0(·1·) . . . , (15)
have E = 1; the same holds true for all kink types (a, b) = (0, ±), (±, 0). Note that kinks/anti-kinks are connected by N = 2 supersymmetry,
Q+: K
a,b→Ka,b. (16)
Multi-kink/anti-kink states can be counted through for-mulas similar to those for the MR state, see eq. (11), for all choices of open chain BC. Followed through to the CFT limit, these counting formulas provide novel expres-sions for characters of the CFT [15].
We now define σ-type bulk defects. These will allow some of the bulk kink states to exist at zero energy (for λ = 0). In the example of eq. (15) this can be done by excluding the configuration ‘11’ at the kink location: this eliminates the anti-kink and turns the kink into an E = 0 state! To treat the ± vacua on equal footing, we repeat the same constraint two steps to the right, and define
σ : . . . λ 1 no 11 λ 1 no 11 . . . , σ0 : . . . 1 λ no 11 1 λ no 11 . . . . (17)
This definition holds for general λ and represents a con-straint in the Hilbert space of states. At λ = 0, a defect σ can connect a vacuum |0i coming in from the left to |+i, |0i or |−i extending to the right, and opposite for
σ0, all at zero energy.
We can now conceive a ‘quantum register’ by taking an open chain, length L = 4l + 1, staggering type λ1 . . . 1λ,
σ-type BC at both ends, and injecting a sequence of 2n
well-separated defects in the order σ0σ . . . σ0σ. This leads
to 3n+1degenerate groundstates at λ = 0, with 2n+1 of
them having the minimal particle number f = 2l − n.
These 2n+1 states form an Ising anyon ‘quantum
regis-ter’. At λ > 0, a unique E = 0 groundstate at f = 2l − n remains, while all other states pair up into doublets at
energies of order e−αLi.
M3 model. -Turning to k = 3 we identify the
follow-ing four PBC groundstates at extreme staggerfollow-ing |1i = . . . 11 ∧100 . . . , | 1 2i = . . . (·1∧· · · ) . . . , |0i = . . . 01 ∧011 . . . , | − 1 2i = . . . 0∧(·11·) . . . , (18)
with ∧ indicating the position of ‘λ’ in the staggering
pattern eq. (4), (·1 · · · ) = 01101 − 01110 + 11001 − 11010 and (·11·) = 1110 − 0111. The energies of the kinks
Ka,b are ma,b = 2|a − b|. This is in agreement with
the kink masses in the N = 2 supersymmetric massive integrable QFT with Chebyshev superpotential W (X) =
1 5X
5− β2X3 + βX [23–26], which we expect to result
from the RG flow set by the staggering perturbation.
On open chains, we define σ1 type BC by excluding
‘111’ near a given edge, and σ2 type by excluding ‘11’.
At the level of the open-chain CFT spectra, a σitype BC
precisely corresponds to the Z3 parafermion spin field
σi: upon changing BC, the various CFT sectors shift
according to the fusion products with these two spin fields. The k = 3 CFT contains, besides a free boson,
Z3 parafermions ψ1,2 and parafermion spin fields σi, ε.
The supercharge Q+ is the zero-mode of the super
cur-rent ψ1V5
2(z). For open/open BC, the M3model realises
the sectors, with m =5
2f − 3 2L − 3 4, {Vm, ψ1Vm, ψ2Vm} (19)
for k = 0, 1, 2 with k ≡ f mod 3. For open/σ1 BC this
becomes, with m =5 2f − 3 2L − 1 4, {σ1Vm, εVm, σ2Vm}, (20)
in accordance with the parafermion fusion rules σ1ψ1= ε
and σ1ψ2= σ2. For open/σ2BC, with m = 52f −32L +14,
{σ2Vm, σ1Vm, εVm}, (21)
in agreement with σ2ψ1 = σ1 and σ2ψ2= ε. The
super-symmetric groundstates are |σ1,2V±1
4i and |V±
3 4i.
Putting σitype BC on both ends, the open-chain CFT
sectors follow the fusion rules σ1σ1 = ψ1+ σ2, σ1σ2 =
1 + ε and σ2σ2 = ψ2+ σ1. As for k = 2, these fusion
rules lead to exponential groundstate degeneracies in the extreme staggering limit. A characteristic case would be
L = 15 sites, σ2-type BC on both ends, and staggering
positioned as ??λ . . .. Here the lowest CFT states are two supersymmetry doublets (at f = 8, 9) at CFT energies
1 1 2 0 -1 2 0 1 2 1 (a) σ2defect 1 1 2 0 -1 2 0 1 2 1 -1 2 (b) σ1 defect
FIG. 3. Fusion rules of bulk defects in the M3 model at
ex-treme staggering.
ECFT = 1/5, ECFT = 4/5, while the λ = 0 M3 model
has four E = 0 vacua. The states connect according to
|1
2iσ2,σ2 ↔ |ψ1V−5/4i, |0iσ2,σ2 ↔ |ψ2V+5/4i,
| −1
2iσ2,σ2 ↔ |σ2V−5/4i, |1iσ2,σ2 ↔ |σ1V+5/4i. (22)
We define bulk defects as
σ1: . . . λ ? ? no 111 . . . , σ2: . . . λ ? ? no 11 . . . , σ0 1: . . . ? ? λ no 111 . . . , σ02: . . . ? ? no 11 λ . . . . (23)
Fig. 3 depicts the corresponding zero-energy fusion rules. They determine the size and structure of the quantum register opened up by an alternating sequence of defects
σi, σj0. In all cases the states at maximal particle number
are made up entirely of vacua |0i and |1i. Restricting the fusion rules to |0i, |1i (drawn lines) gives Fibonacci number degeneracies. The N = 2 supersymmetry acts
within the register, with Q− mapping the |0i, |1i into
linear combinations of |1
2i, | −
1
2i. In the example of L =
5l sites, σ2/σ2BC, staggering type ? ? λ . . . and n σ2/σ02
-type defects, the degeneracy at f = 3l is Fibon+3, with
Fiboj = 1, 1, 2, 3, . . .. Eq. (22) is the case n = 0. For
n = 1 one finds 3, 4, 1 states at f = 3l, . . . , 3l − 2, n = 2 gives 5, 9, 5, 1 states at f = 3l, . . . , 3l − 3, etc.
For the general Mk model, defects eliminating k + 1 − j
consecutive ‘1’s will correspond to the Zk parafermion
spin fields σj, j = 1, . . . , k − 1 [15].
Conclusions. - We have demonstrated that
(bound-ary and bulk) defects in the Mkmodels off-criticality lead
to quantum registers that are protected by supersymme-try. It will be interesting to explore ways to manipulate these registers, so as to act on the quantum informa-tion that can be stored in the supersymmetry protected ground states. One idea is that of a 1D braid protocol (as in [9]); one expects that this will result in the braid
ma-trices as they are known for the corresponding 2D (RRk)
topological phases. It will also be interesting to see if a
dual formulation of the Mk models, with 1D topological
order rather than order of CDW type, can be obtained. One expects that operators that preserve supersymmetry
in the Mk models correspond to operators that are local
in the dual variables of the topological phase, and are thus unable to split the exponential degeneracies.
Many extensions of the ideas presented here are
feasi-ble. In the M2model, alternative bulk defects, based on
a ‘no 0’ rather than a ‘no 11’ condition, lead to Fibonacci
number degeneracies. The M1 model on a square ladder
is in many ways similar to the M2 model [27] and it is
natural to introduce σ-type defects in the 2D M1 model
on the octagon-square and square lattices [28, 29]. In all these cases, we expect non-trivial fusion relations to emerge.
We thank Erez Berg, Tarun Grover, Yingfei Gu, Chris-tian Hagendorf, Liza Huijse, Steve Simon, and, in par-ticular, Paul Fendley and Ville Lahtinen, for discussions, and the INFN for hospitality in Firenze, where part of this work was done. T.F. is supported by the Netherlands Organisation for Scientific Research (NWO).
∗ t.b.fokkema@uva.nl †
c.j.m.schoutens@uva.nl
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